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EFFICIENT RESPONSE SURFACE APPROACH FORRELIABILITY ESTIMATION OF COMPOSITE STRUCTURES

M. Rais-Rohani*, M. N. Singh**

* Professor, Senior Member AIAA, masoud@ae.msstate.edu ** Graduate Research Assistant

Department of Aerospace EngineeringMississippi State University, Mississippi State, MS 39762

Abstract

In assessment of component reliability andreliability-based structural optimization, efficientevaluation of reliability index is of considerableimportance. In this paper, we will examine thefeasibility of Gauss quadrature points associated withnumerical integration of multivariable functions toperform targeted sampling of the design space and rapidcalculation of regression coefficients in first- andsecond-order response surface models of failurefunctions. Parametric uncertainty is considered byprobabilistic modeling of design parameters. Variousalternative strategies for approximation of componentreliability index are considered with application to twostructural components made of composite materials.Design sensitivity analysis is performed to measure theinfluence of each random variable on the estimatedreliability index. The advantages and disadvantages ofeach strategy are identified and the one approachconsidered the most efficient in terms of computationalrequirements and accuracy is identified.

Introduction

The desire to increase structural efficiency whilemaximizing reliability and robustness has fueled thegrowth of non-deterministic approaches during the pastdecade and has led to the development and application ofmany techniques under the general category ofreliability-based structural optimization (RBSO). For ageneral description of RBSO and related research, thereader should refer to Frangopol.1

Many approximate techniques have been developedto assess the structural reliability of a component or asystem, which can be broadly categorized into twogroups: a) random sampling methods, and b) analyticalmethods.

In all cases, the non-cognitive uncertaintiesassociated with modeling, material, loading, etc. arecaptured using a variety of techniques that rely onprobabilistic treatment of random variables. In contrast,

cognitive uncertainties are modeled primarily usingfuzzy sets theory.2

Prevalent among the random sampling methods isthe traditional Monte Carlo simulation and its variantssuch as adaptive importance sampling3 and robustimportance sampling4, which have been developed toreduce the sampling requirement by adjusting thesampling domain to the most important regions.

Analytical methods include the first- and second-order reliability methods (FORM and SORM). Thesemethods rely on explicit or implicit formulation of thelimit-state or failure function in terms of basic randomvariables, and the calculation of the reliability indexsuch as that proposed by Hasofer and Lind5.

Depending upon the complexity of failure analysis,both simulation based and analytical reliability methodscan be very time consuming. To enhance the efficiencyof reliability analysis many approaches and strategieshave been proposed, some of which rely on the use ofresponse surface methodology (RSM). For example,Bucher and Bourgund6 used an adaptive interpolationscheme to represent the system behavior by an RSmodel, which was then combined with importancesampling to obtain the desired reliability estimates. Liuand Moses7 combined the sequential RSM with MonteCarlo importance sampling to develop a reliabilityanalysis program for aircraft structural systems. Liawand DeVries8 developed a reliability-based optimaldesign process by integrating reliability and variabilityanalysis with optimization design processes using theresponse surface approach.

It is evident from the literature that the use of RSmodels in lieu of "exact" analysis procedure cansignificantly reduce the computational cost for a singlestructural analysis. However, accurate estimation of theunknown coefficients in an RS model by commonlyused techniques (e.g., Monte Carlo simulation, centralcomposite design) usually require a very largepopulation of response samples. Hence, any promise ofcomputational savings offered by RSM could rapidlydiminish if the population size is too large with each

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization4-6 September 2002, Atlanta, Georgia

AIAA 2002-5604

Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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response sample obtained from a computationallyintensive analysis procedure.

In this paper, we will examine the use of Gaussquadrature points associated with numerical integrationof multivariable functions9,10 to perform "targeted"sampling of design space. The response observationsobtained through evaluation of complex failurefunctions at multiple target points are used to developcorresponding first- and second-order response surfacemodels.

Within the context of reliability-based designoptimization, alternative strategies for estimation ofreliability index are presented and applied to twoexample problems, one on ply failure in amultidirectional composite laminate and the other onaxial buckling of a composite circular cylinder. Thecomputational efficiency of each strategy and accuracyof the estimated reliability index are examined with thehelp of these two example problems.

In addition to calculating the reliability index, itsprobabilistic sensitivity derivatives are evaluated foreach example problem to identify influential parametersand determine the effect of parametric uncertainty ineach case.

Reliability-Based Design Optimization

In reliability-based design optimization of structuralsystems involving multiple modes of failure, theoptimization problem is stated in the following form:Find the vector of design variables Y which

Min. f (X )

(1)S.T. i

e ≥ ie min , i = 1,2,...,NF

s ≥ smin

Yjl ≤ Yj ≤ Yj

u , j = 1,2,...,NDV

where the objective function depends on the vector ofrandom variables, X . Design constraints include limitson minimum reliability index associated with eachfailure mode denoted by i

e as well as system reliability

index denoted by s . Design variable vector

(Yj , j = 1,2,...,NDV ) is a subset of random variable

vector X with specified lower and upper bounds or sideconstraints.

Our focus in this paper is not to solve Eq. (1) for aparticular problem rather we intend to show an efficientapproach for calculation of component reliability index

ie which is a crucial part of any reliability-based

design optimization problem.

Reliability Index

Calculation of reliability index begins with theformulation of a failure (or limit-state) function, whichmay be expressed, in general, as g( X1, X2,...,Xn ) where

n denotes the number of random variables. A failurefunction identifies an n-dimensional surface separatingthe failure region characterized by g ≤ 0 from the safe

region characterized by g > 0 . For computational

efficiency, the probability of failure defined asPf = P(g ≤ 0) is estimated indirectly through the

calculation of reliability index .

When failure function involves normally distributedand uncorrelated random variables, the Hasofer-Lindreliability index based on first order reliability method(FORM) is found as11

HL = u*T u*( )1/2= −

ui*

i =1

n

∑ g

U i*

g

Ui*

2

i =1

n

∑ (2)

where HL represents the minimum distance from the

origin of the reduced coordinate system (u-space) to thetangent hyperplane of the failure surface (g = 0 ) at the

design point u* = u1*, u2

*,...,un*( )T

. This is also the point

where the partial derivatives of limit-state function( g / U i

* ) are evaluated. The probability of failure

Pf = P(g ≤ 0) is related to reliability index as

Pf = Φ (− HL ) , where Φ is the cumulative distribution

function of a standard normal variate. Finding u* and HL based on Eq. (2) becomes an

iterative process requiring a solution sequence such asthat developed by Lee12 and described in the appendix.

The sensitivity factors representing theprobabilistic sensitivity derivatives of with respect tothe mean value X i

and standard deviation ˜ X i

are

calculated (at the design point) as13

Xi

= −ui

*

˜ X i

(3)

˜ Xi

= −ui

*2

˜ X i

(4)

While the sensitivity factors in Eq. (3) quantify theinfluence of each random variable on , those in Eq. (4)

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quantify the effect of uncertainty in each randomvariable on the reliability index.

Although the iterative procedure for finding , asdescribed in the appendix, is simple in principle, it canbecome computationally intensive if the limit-stateequation is not an explicit function of random variablesor the procedure for implicit evaluation of limit-statefunction and its derivatives ( g / U i

* ) requires CPU-

intensive analyses.To remedy this problem, we investigated the use of

response surface methodology (RSM) for fast andaccurate representation of structural behavior byresponse surface (RS) models of limit-state functions asdiscussed next.

Response Surface Modeling of Failure Functions

RSM is used to develop an algebraic responsesurface equation relating the structural response ofinterest to the primary variables that influence it. Inproblems involving more than one primary variable, amultiple linear regression model is used to estimate thestructural response. The two practical options forestimating the limit-state functions involve using first-or second-order RS models defined as

ˆ g (X ) = a + biXii=1

n

∑ (5)

ˆ g (X ) = a + biXii=1

n

∑ + ciXi2

i =1

n

∑ (6)

Equation (5) involves n+1 unknown regressioncoefficients while Eq. (6), with the interaction terms( XiX j , i ≠ j ) ignored, involves 2n+1 unknown

coefficients. In both cases, the coefficients areestimated using the least-squares technique based on apopulation of response samples.

The two important considerations in application ofRSM are: 1) deciding on what order RS model to use,and 2) finding the regression coefficients with areasonable degree of accuracy and efficiency.

First-order RS models are commonly used when thehigher-order effects are small or nonexistent. Second-order models are used when the non-linear terms have asignificant influence on the accuracy of the model. Inmany problems, the interaction terms have minimaleffect on the response and are usually ignored.

The unknown regression coefficients in an RSmodel are estimated using the least squares techniquebased on a population of response samples. Commonlyused techniques for generating the necessary response

data include direct Monte Carlo simulation and centralcomposite design. However, in both cases, a very largenumber of response samples need to be generated. Forexample, using a central composite design for a modelinvolving n variables requires 2n + 2n +1 responsesamples, which may rapidly diminish any potentialcomputational savings offered by RSM as the value ofn is increased.

In this paper, we have examined an alternativeprocedure for reducing the number of response samplesneeded for accurate estimation of regression coefficients.This procedure is based on the formulas developed byZhou and Nowak10 for numerical integration of jointlydistributed random vectors. Their technique involvesfinding appropriate non-uniformly spaced quadraturepoints and weights for a non-product point integrationof functions or random variables. We have used Zhouand Nowak's choice of quadrature points to define thevalue of ith random variable for jth sampling experimentaccording to the formula

xi j= Xi

+ zij˜

Xii = 1,2,...,n (7)

j = 1,2,...,m

The transformation of random variables from standardnormal to basic random space depends on therelationship between n and m . For m = n+1, thequadrature points are defined10 as Zj = (z1 j

, z2 j,...,zn j

) ,

where

Z1 = ( n ,0,0,...,0)

Z2 = −1

n,

n + 1( ) n − 1( )n

,0,...,0

Z3 = −1

n, −

(n + 1)

n(n − 1),

n + 1( ) n − 2( )(n − 1)

,0,...,0

M (8)

Zn = − 1

n, − (n + 1)

n(n − 1), − (n + 1)

(n −1)(n − 2),

...,n + 1( )

2

Zn +1 = − 1

n, − (n + 1)

n(n − 1), − (n +1)

(n − 1)(n − 2),

...,−(n + 1)

2

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Hence, in the first simulation all random variables arekept at their respective mean values except for the firstone that is perturbed by x1 = X1

+ n ˜ X1

.

For m = 2n, the quadrature points are defined10 as

Z1 = − Zn+1 = n ,0,...,0( )Z2 = − Zn+ 2 = 0, n ,...,0( )M

Zn = − Zn+ n = 0,0,..., n( )

(9)

Next, we will consider various strategies forestimation of reliability index.

Strategies for Estimation of Reliability Index

To investigate the accuracy and efficiency of usingtargeted sampling technique based on Gauss quadraturepoints, we examined six different cases described asfollows:

Case 1: Calculate the reliability index directly froma Monte Carlo simulation based on "exact" failurefunction evaluation at each cycle. The probability offailure based on M simulation cycles is determined asPf = M f / M , where M f is the number of simulations

resulting in ply failure. Reliability index is found using=−Φ −1 (Pf ).

Case 2: Calculate the reliability index from Eq. (2)using the iterative procedure described in the appendix.Evaluate the partial derivative of limit-state functionwith respect to each random variable ( g / X i ) using

forward finite-difference scheme. Find an appropriatestep size parameter for each random variable (i.e.,∆Xi = Xi ) that would provide reasonably accurate

derivatives.Case 3: Estimate the reliability index by using a

first-order RS model to approximate the failurefunction. In this case, the n+1 regression coefficientsare found using m = n+1 quadrature points defined byEq. (8). Because the limit-state function isapproximated by a first-order RS model, reliabilityindex can be found directly using

ˆ =a + bi

i =1

n

∑ X i

bi˜

X i( )2

i =1

n

∑ (10)

Case 4: Estimate the reliability index by using afirst-order RS model (as in Case 3), but use m = 2n

replications, as in Eq. (9), for estimation of regressioncoefficients.

Case 5: Estimate the reliability index by using asecond-order RS model of failure function. In this case,the 2n+1 regression coefficients are found using m = 2nquadrature points defined by Eq. (9) plusZ2n +1 = 0,....,0( ) . Hence, 2n+1 replicates or response

samples are used to estimate 2n+1 regressioncoefficients. Using the procedure described in the

appendix, we determine ˆ . The difference between this

case and Case 2 is that here the limit-state function isapproximated by a second-order RS model.Furthermore, the derivatives of limit-state function arefound directly from analytical differentiation of the RSequation. It is also important to point out that during

the iteration process for ˆ , the coefficients of the RS

model are not recalculated.Case 6: Estimate the reliability index similar to

that in Case 5. However, while the derivatives of limitstate function are obtained from the second-order RSmodel, use the exact evaluation of limit-state function

for g(x*) (see the appendix) at each iteration step forˆ .

The utility and accuracy of these strategies areexamined with the help of following example problems.

Example 1: Ply Failure in a Composite Laminate

In this example, we considered a thin, symmetricmulti-layered composite laminate in a state of planestress with all laminae perfectly bonded together. Usingthe laminated plate theory and considering laminatesymmetry and no bending and twisting moments, thein-plane stresses in principal material directions (PMD)in the k th ply are found as

{ }k = T[ ]k Q [ ]k

A[ ]−1 N{ } (11)

where { }kT = { 1, 2 , 12}k

T . [T ]k and [Q ]k are the

transformation and the transformed reduced stiffnessmatrices of k th ply, respectively. [A] is the laminateextension stiffness matrix, and N{ } is the stress

resultant vector such that {N}T = {Nx , Ny , N xy}T .

Using Tsai-Wu criterion for ply failure defined as

f1 1 + f2 2 + f11 12 + f22 2

2 + f66 122 + 2 f12 1 2 = 1

(12)where

f1 =1

F1t −

1

F1c , f11 =

1

F1tF1

c , f2 =1

F2t −

1

F2c ,

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f22 =1

F2tF2

c , f66 =1

F122 , f12 = −

1

2

1

F1tF1

cF2tF2

c ,

we obtain a failure function in the form

gpfk = 1− ( f1 1 + f2 2 + f11 1

2 + f22 22 +

f66 122 + 2 f12 1 2 )k

(13)

Equation (13) is an implicit function of randomvariables as defined in Table 1.

A preliminary analysis was performed on three 16-ply thick laminates with ply patterns C1:

±45/90 4 / m45( )s; C2: ±45/0 4 / m45( )

s; and C3:

±45/0 / 0 / 90/90 / m45( )s .

Table 1 shows the list of probabilistic randomparameters and associated uncertainties. The data onmaterial properties (mean and standard deviation) areobtained from MIL-HDBK-17-2E. All random variablesare assumed normally distributed.

Based on Tsai-Wu failure criterion for loadingcondition {N}T = {Nx ,0,0}T , the ±45 and 90-degree

layers are weaker and more likely to fail if the laminateis placed in axial tension while the 0-degree layers areweaker and more likely to fail if the laminate is placedin axial compression. In the case of C1 and C3laminates, the 90-degree ply group is the first to fail ata load of Nx = 1,352 lb/in . and Nx = 3,099 lb/ in .,

respectively. In the case of C2 laminate, the 0-degreeply group is the first to fail at Nx = − 5,383 lb/in .

Comparison of Results for Cases 1 through 6

To compare the results for Cases 1 - 6, we usedlaminate C1 under a deterministic axial tensile load Nx .

We determined the reliability index associated withfailure in the 90-degree ply group, which would be thefirst group to fail under these conditions. For laminateC1, the problem involves a total of 15 normal anduncorrelated random variables.

The results in each case for different values ofapplied load are given in Table 2 below. For Cases 2,

5, and 6 involving an iterative solution for ˆ ,

ˆ g ≤ 1x10−3 and | ˆ k +1 − ˆ

k | ≤ 1x10−4 (k denotes

iteration number) are used as hard and soft convergencecriteria, respectively. For Case 1, the results are basedon 3000 simulation cycles or 3000 exact calculations offailure function in Eq. (13). For Case 2, the number ofexact failure function evaluations is represented byk(n+1) (n denotes the number of random variables)requiring n+1 function calls for calculation of partial

derivatives ( g / U i* ) at each iteration. For the results

in Table 2, k = 3, 4, 5, 6 for Nx = 1200, 1000, 800,700 lb/in., respectively.

Table 1 Statistical properties of random variablesVariable Distribution Mean Value C.O.V.

E1, psi Normal 18.0E+06 0.032E2, psi Normal 1.35E+06 0.043

12 Normal 0.226 0.050G12, psi Normal 0.543E+06 0.052F1

t , psi Normal 258E+03 0.098

F1c , psi Normal 204E+03 0.065

F2t , psi Normal 7.76E+03 0.107

F2c , psi Normal 34.6E+03 0.095

F12 , psi Normal 14.8E+03 0.032

tply, in. Normal 0.005 0.050

ply, deg. Normal ±45, 0, 90 5.00a

aStandard Deviation

For Case 3, the number of exact failure functionevaluations is n+1 or 16. For Case 4, the number ofexact failure function evaluations is 2n or 30. For Case5, the number of exact failure function evaluations is 2n+ 1 or 31. And finally, for Case 6, the number of exactfailure function evaluations is 2n+1+k. For the resultsin Table 2, k = 4, 8, 15, 24 for Nx = 1200, 1000, 800,700 lb/in., respectively. The total number of exactfunction (i.e., Eq. (13)) evaluations for each case andload combination is shown in Table 3.

Table 2 Estimated values of reliability index ˆ for the

90-degree ply group in laminate C1 for Cases 1 - 6Case

Nx ,

lb/in

1 2 3 4 5 6

1200 0.54 0.60 0.52 0.48 0.60 0.601000 1.45 1.52 1.74 1.63 1.65 1.53800 2.56 2.66 3.68 3.40 3.10 2.67700 3.21 3.33 5.11 4.70 4.07 3.35

The trends observed in Table 2 indicate that as thereliability index increases the results from Cases 3 and4, based on a first-order RS model, tend to deviatesignificantly from those found using Monte Carlosimulation (MCS). By contrast, the results for Cases 2and 6 agree reasonably well with those of MCS at aconsiderably less computational cost. Of all thestrategies considered, the approach based on Case 6appears to be the most promising. Comparing Cases 1,2, and 6, we find the results of Case 2 to be closer to

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Case 1, but by a relatively insignificant amount.However, when considering the number of exactfunction evaluations as shown in Table 3, it is clearthat the approach in Case 6 is considerably moreefficient than the other two.

Table 3 Number of exact ply failure functionevaluations in Cases 1 - 6

Case

Nx ,

lb/in

1 2 3 4 5 6

1200 3000 48 16 30 31 351000 3000 64 16 30 31 39800 3000 80 16 30 31 46700 3000 96 16 30 31 55

The values of basic random variables at the designpoint or MPP for Nx = 700 lb/in. in Cases 2 and 6 arecompared in Table 4. The MPP values represent theworst combination of design properties resulting in plyfailure.

Table 4 Comparison of mean and MPP values for Cases2 and 6 in Table 2 at Nx = 700 lb/in.

RandomVariable, Xi

MeanValue

MPP ValueCase 2 Case 6

E1, E+06 psi 18.0 17.89 17.88E2, E+06 psi 1.35 1.371 1.370

12 0.226 0.22565 0.22571G12, E+06 psi 0.543 0.539 0.541F1

t , E+03 psi 258 259.14 260.06

F1c , E+03 psi 204 203.15 202.86

F2t , E+03 psi 7.76 5.86 5.72

F2c , E+03 psi 34.6 34.58 35.01

F12 , E+03 psi 14.8 14.80 14.80

t45 , in. 0.02 0.01976 0.01980

t−45 , in. 0.02 0.01976 0.01980

t90 , in. 0.04 0.03927 0.03959

45 , deg. 45 53.243 52.784

−45 , deg. -45 -53.243 -52.784

90 , deg. 90 90.0021 90.0019

Sensitivity Analysis

The probabilistic sensitivity derivatives arecalculated using Eqs. (3) and (4) and are normalized as( / X i

)n = ( / Xi)( X i

/ ) and

( / ˜ X i

)n = ( / ˜ Xi

)( ˜ X i

/ ) to eliminate the

scaling effect of random variables. The original andnormalized derivatives for Case 2 are shown in Table 5.Of all random variables considered, failure strength F2

t

is found to have the greatest influence, followed byorientation angle of the ±45-degree plies, and elasticitymoduli E2, and E1. The effect of orientation angle of±45 plies in the failure of 90-degree ply group is due tothe presence of in-plane laminate stiffness matrix in Eq.(11).

Table 5 Probabilistic sensitivity derivatives of ˆ for

the 90-degree ply group in laminate C1 for Case 2 at Nx

= 700 lb/in.

RandomVariable,

Xi

ˆ

Xi

ˆ

X i

n

ˆ

˜ Xi

ˆ

˜ X i

n

E1 1.01E-07 0.546 -1.95E-08 -0 .0034E2 -1.92E-06 -0 .777 -7.05E-07 -0 .0123

12 8.29E-01 0.056 -2.59E-02 -0.0001G12 1.52E-06 0.248 -2.16E-07 0.0000F1

t -5.34E-07 -0.041 -2.41E-08 -0.0002

F1c 1.48E-06 0.091 -9.58E-08 -0.0004

F2t 8.25E-04 1.924 -1.89E-03 -0 .4700

F2c 6.45E-07 0.007 -4.57E-09 0.0000

F12 0.00E+00 0.000 0.00E+00 0.0000t±45 7.23E+01 0.109 -1.74E+01 -0.0013t90 5.50E+01 0.083 -2.01E+01 -0.0015±45 -9.90E-02 -1 .337 -1.63E-01 -0 .245090 -2.53E-05 -0.001 -1.07E-08 0.0000

As for the effect of parametric uncertainty,represented by the normalized derivatives of reliabilityindex with respect to standard deviation shown in Table5, we find the uncertainty in the same random variablesmentioned earlier to be of most concern. The negativesign indicates that as the uncertainty is reduced thereliability index is increased.

In Table 6, the normalized sensitivity derivativesobtained in each case are tabulated for comparison.Only the most important derivatives identified in Table5 are considered for this comparison. The reliabilityindex used for normalization of each derivative is thatfound for the corresponding case as listed in Table 2.

The best overall agreement is between Cases 2 and6. For E1 and E2, the derivatives from the linear RSmodel in Case 4 are much closer to those in Case 2than those found in Cases 3 and 5. The normalizedsensitivity derivatives ( / ˜

X i)n showed a similar

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trend to that seen in Table 6, hence, are not presentedhere.

Table 6 Normalized sensitivity derivatives ( ˆ / X i)n

for the 90-degree ply group in laminate C1 for Cases 2 -6 at Nx = 700 lb/in.

Case

Xi 2 3 4 5 6

E1 0.546 0.693 0.556 0.468 0.600E2 -0.777 -0.517 -0.684 -0.566 -0.728F2

t 1.924 0.826 1.254 1.728 2.045

±45a -1.337 -1.206 -1.010 -0.998 -1.250

a Value shown for Case 3 is for +45 layer. For the -45 layer,normalized derivative is -0.851

It is worth noting that even a simple linear modelas in Cases 3 and 4 is capable of providing usefulsensitivity information to identify the influentialvariables affecting component reliability. Hence, it isreasonable to use a simple first-order RS model basedon n+1 or 2n response samples to quickly identify theimportant variables, and then use the more accuratemodel as in Case 6 for subsequent design optimization.

Example 2: Axial Buckling of a Composite Cylinder

The problem of interest is a thin-walled compositecircular cylinder under a uniform axial compression.The nominal diameter and length are 10 in. and 20 in.,respectively. The cylinder is made of a symmetriclaminate of multidirectional layers made of graphite-epoxy composite material. The ply pattern considered inthis example is the same as that defined by laminate C1in previous example. The applied load acting on thecylinder is treated as uniform and deterministic, and theloaded edges are assumed perfectly clamped. Thematerial properties are the same as those definedpreviously in Table 1.

For axial buckling analysis, the displacement fieldis described by the first-order shear deformation theorywhile the strain-displacement relations are based onSanders-Koiter shell theory. The midplanedisplacements and rotations are approximated bydifferent Ritz series formulated in terms of 12th degreeLegendre polynomials. Following the formulation ofelastic strain energy and the work done by the appliedload, the principle of minimum total potential energy isapplied and the resulting eigenvalue problem is solvedfor the critical buckling load factor cr such that

Ncr = cr N (14)

where N is the uniformly-distributed applied axial load.A computer implementation of the described analysisprocedure14 is used to calculate the axial buckling loadsfor circular cylinders with multidirectional laminateconfigurations.

The limit-state function for axial buckling may beexpressed as

gb =Ncr

Na

− 1 (15)

Similar to Eq. (13) in Example 1, Eq. (15) is alsoan implicit function of random variables (see Table 1).However, unlike Example 1, the exact calculation oflimit-state function involves a rather time consumingeigenvalue analysis for cr . Therefore, in the context

of reliability-based optimization where reliability indexand derivatives of limit-state function are calculatedrepeatedly, computational time can rapidly grow beyondwhat is considered practical. Hence, in this example,only Cases 2 through 6 are considered with the solutionresults compared for various applied loads as discussednext.

Comparison of Results for Cases 2 through 6

Similar to the case of ply failure, we also examinedthe efficiency of each strategy and accuracy of reliabilityindex estimates for buckling failure of the compositecylinder.

In this case, the "exact" buckling failure functionrefers to Eq. (15) with the buckling load found directlyfrom the shell analysis code14. The first- and second-order RS models are similar to those used for ply failureas defined by general Eqs. (5) and (6). However, the listof random variables used for buckling failure includescylinder length and diameter and excludes ply tension,compression, and shear strength properties (see Table1). Hence, the buckling failure function is a function ofn = 10 normal and uncorrelated random variables.

The results obtained from Cases 2 through 6 areshown in Table 7 below. Among Cases 3 - 6,involving the use of RS models, Case 6 provides theclosest answers to those in Case 2. Unlike in the caseof ply failure, Case 3 with the first-order RS model ofbuckling limit-state function consistently

underestimates the value of ˆ as compared to the other

cases. Also, the results in Case 4 with a first-order RSmodel are slightly better than those in Case 5 with asecond-order RS model. In comparing Cases 5 and 6,we notice that the reliability index estimates improve asa result of calculating the limit-state function "exactly"

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by calling the shell code at each iteration step for ˆ

(see appendix) while estimating the derivatives of limitstate function from a corresponding second-order RSmodel.

Table 7 Estimated values of ˆ for axial buckling in

Cases 2 - 6Case

Nx , lb/in 2 3 4 5 6

-4000 0.60 0.51 0.55 0.68 0.62-3500 1.74 1.61 1.77 1.87 1.79-3000 3.04 2.71 2.99 3.01 3.09-2500 4.58 3.80 4.21 4.09 4.75

Table 8 gives an indication of the computationalcosts for reliability analysis in Cases 2 - 6. With only11 and at least 33 exact buckling failure functionevaluations, respectively, Cases 2 and 3 are at theopposite ends of scale for computational cost. Also,the difference between Cases 5 and 6 is not as large asthat seen in Table 3 for ply failure. The mostimportant comparison is once again between Cases 2and 6. It is clear that the technique used in Case 6 issubstantially more efficient than that used in Case 2.

Table 8 Number of exact buckling failure functionevaluations in Cases 2 - 6

Case

Nx , lb/in 2 3 4 5 6

-4000 33 11 20 21 24-3500 33 11 20 21 25-3000 44 11 20 21 27-2500 55 11 20 21 27

The values of basic random variables at the designpoint or MPP for Nx = -3000 lb/in. found in Cases 2and 6 are compared in Table 9. The MPP valuesrepresent the worst combination of design propertiesresulting in buckling failure of the cylinder.

Sensitivity Analysis

The reliability sensitivity analysis is performed forCases 2 - 6 with results for Case 2 shown in Table 10.Of all random variables considered, cylinder diameter,elasticity modulus E1, and ply angles have the greatestinfluence on cylinder buckling reliability index.

Table 9 Comparison of mean and MPP values for Cases2 and 6 in Table 7 at Nx = -3000 lb/in.

RandomVariable, Xi

MeanValue

MPP ValueCase 2 Case 6

E1, E+06 psi 18.0 17.68 17.58E2, E+06 psi 1.35 13.36 13.40

12 0.226 0.22585 0.22586G12, E+06 psi 0.543 0.5389 0.5396

t±45 , in. 0.04 0.0372 0.0371

t90 , in. 0.04 0.0375 0.0375

±45 , deg. 45 54.44 54.81

90 , deg. 90 85.33 88.15

D, in. 10 10.455 10.547L, in. 20 20.23 20.25

In Table 11, the normalized sensitivity derivativesobtained in each case are tabulated for comparison.Only the most important derivatives identified in Table10 are considered for this comparison. The reliabilityindex used for normalization of each derivative is thatfound for the corresponding case as listed in Table 7.

Table 10 Probabilistic sensitivity derivatives of ˆ for

axial buckling in Case 2 at Nx = -3000 lb/in.

RandomVariable,

Xi

ˆ

Xi

ˆ

X i

n

ˆ

˜ Xi

ˆ

˜ X i

n

E1 3.16E-07 1.872 -1.74E-07 -0 .0330

E2 1.42E-06 0.628 -3.50E-07 -0.0067

12 0.3872 0.029 -5.15E-03 0.0000

G12 1.72E-06 0.307 -2.51E-07 0.0000

t±45 230.0 0.378 -321.17 -0.0264

t90 206.0 0.339 -258.15 -0.0212

±45 -0.1243 -1 .840 -0.2347 -0 .3860

90 0.0615 1.821 -0.0574 -0 .0945

D -0.5998 -1 .973 -0.5462 -0 .0898

L -0.0757 -0.498 -1.74E-02 -0.0057

For discussion of reliability-based optimization ofcomposite cylinders involving the use of responsesurface modeling, similar to that in Case 3, refer to ref.[15] by Su et al.

American Institute of Aeronautics and Astronautics

9

Table 11 Normalized sensitivity derivatives

( ˆ / X i)n for axial buckling in Cases 2 - 6 at Nx = -

3000 lb/in.Case

Xi 2 3 4 5 6

E1 1.872 3.092 2.720 2.470 2.390±45 -1.840 -1.608 -1.419 -1.888 -1.85090 1.821 -0.874 0.299 0.638 0.696D -1.973 -2.604 -2.959 -2.374 -2.294

Summary and Conclusions

In this paper, we examined the feasibility of Gaussquadrature points associated with numerical integrationof multivariable functions to perform targeted samplingof the design space and rapid calculation of regressioncoefficients in first- and second-order response surfacemodels of failure functions. We analyzed alternativestrategies for approximation of component reliabilityindex. We applied these strategies to two exampleproblems to investigate the effects of parametric andmodeling uncertainties on reliability of compositestructures. We also performed sensitivity analysis tomeasure the influence of each random variable on theestimated reliability index.

Among the strategies considered, the approachinvolving the use of exact limit-state functionevaluation at each iteration point and the use of second-order response surface model for calculation of analyticalpartial derivatives (i.e., Case 6) proved to be the mostefficient.

In all cases, the sensitivity derivatives of reliabilityindex with respect to the standard deviation provided agood measure of the effect of parametric uncertainty.

Acknowledgements

This research was funded in part by the NASALangley Research Center under grant NAG 1-02099 andby a Honda Fellowship at Mississippi State University.

References

1. Frangopol, D. M., "Reliability-Based OptimumStructural Design," in Probabilistic StructuralMechanics Handbook, Theory and IndustrialApplications , Edited by C. Sundararajan, Chapmanand Hall, 1995.

2. Ayyub, B. M. and Chao, R.-J., "UncertaintyModeling in Civil Engineering with Structural andReliability Applications," in Uncertainty Modeling

and Analysis in Civil Engineering , Edited by B. M.Ayyub, CRC Press, 1998.

3. Wu, Y.-T., "Computational Methods for EfficientStructural Reliability and Reliability SensitivityAnalysis," AIAA Journal, Vol. 32, No. 8, 1994,pp. 1717-1723.

4. Torng, T.Y., Lin, H.-Z., and Khalessi, M.R.,"Reliability Calculation Based on a RobustImportance Sampling Method," Proceedings of the37th AIAA/ASME/ASCE/AHS/ASC Structures,Structural Dynamics, and Materials Conference,Salt Lake City, UT, Apr. 15-17 1996. Part 3, pp.1316-1325.

5. Hasofer, A.M. and Lind, N., "An Exact andInvariant First-Order Reliability Format," Journalof Engineering Mechanics, Vol. 100, No. EM1,1974, pp. 111-121.

6. Bucher, C.G. and Bourgund, U., "A Fast andEfficient Response Surface Approach for StructuralReliability Problems," Structural Safety, Vol. 7,No. 1, 1990, pp. 57-66.

7. Liu, Y. W. and Moses, F., "A Sequential ResponseSurface Method and its Application in theReliability Analysis of Aircraft StructuralSystems," Structural Safety, Vol. 16, No. 10,1994, pp. 39-46.

8. Liaw, L. D. and DeVries, R. I., "Reliability-BasedOptimization for Robust Design," InternationalJournal of Vehicle Design, Vol. 25, Nos.1/2,2001, pp. 64-77.

9. Stroud, A. H. and Secrest, D., "ApproximateIntegration Formulas for Certain SphericallySymmetric Regions," Mathematics ofComputation , Vol. 17, 1963, pp. 105-135.

10. Zhou, J-H. and Nowak, A., "Integration Formulasto Evaluate Functions of Random Variables,"Structural Safety, Vol. 5, 1988, pp. 267-284.

11. Shinozuka, M., "Basic Analysis of StructuralSafety," Journal of Structural Engineering, Vol.109, No. 3, 1983, pp. 721-740.

12. Lee, Y.-H., "Stochastic Finite Element Analysis ofStructural Plain Concrete," Ph.D. Dissertation,Department of Civil, Environmental andArchitectural Engineering, University of Colorado,Boulder, CO., 1994.

13. Madson, H. O., Krenk, S., and Lind, N. C.,Methods of Structural Safety , Prentice-Hall, Inc.,New Jersey, 1986, pp. 120-123.

14. Jaunky, N. and Knight, N., "An Assessment ofShell Theories for Buckling of Cylindrical Panels,"International Journal of Solids and Structures,Vol.36, 1999, pp. 3799-3820.

American Institute of Aeronautics and Astronautics

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15. Su, B., Rais-Rohani, M., and Singh, M. N.,"Reliability-Based Optimization of AnisotropicCylindrical Shells with Response SurfaceApproximations of Buckling Instability,"Proceedings of the 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, andMaterials Conference, Denver, CO, Apr. 22-25,2002, AIAA Paper No. 2001-1386.

Appendix

The algorithm developed by Lee12 for finding thereliability index is known as the direct derivation

approach. Starting from an initial estimate for the mostprobable failure point, the reliability index isapproximated as

k +1 = k +g k x( )

g

Ui

k

2

i =1

n

∑

(A-1)

where k refers to iteration number. It is common tostart the iteration at mean values of random variablesx1 = E(x ) with 1 = 0 . The partial derivatives of the

limit-state function with respect to random variables inreduced coordinate system (u-space) are found as

g

Ui

=g

Xi

˜ Xi

(A-2)

where ˜ X i

denotes the standard deviation of random

variable Xi in basic coordinate system (x-space) with

mean X isuch that

Ui =Xi − X i

˜ Xi

i = 1,2,...,n (A-3)

Next, the updated location uik +1 in reduced coordinate

system is found as

uik +1 = − i

k k +1 (A-4)

where i is the direction cosine obtained using

ik =

g

Ui

k

g

Ui

k

2

i =1

n

∑

(A-5)

The updated position in x-space is found as

xik +1 = X i

+ uik +1 ˜

X i (A-6)

Steps described by Eqs. (A-1) - (A-6) are repeated until

convergence is reached (i.e., g x*( ) ≤ , = 1x10−3 ).

Then the converged value of represents the shortest

distance from the origin of the reduced coordinatesystem to the failure surface such that

ui* = − i

* (A-7)

The relationship between and limit-state surface in

the case of a 2-dimensional problem is depicted in figureA-1.

Fig. A-1 Graphical representation of failure function,design point and reliability index

g(U) > 0(safe region)

U1

U2

u*

g(U) = 0

g(U) < 0(failure region)

design point